Abstract
In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.
Highlights
A special type of initial value problems, treated in this work, involves highly oscillatory systems and problems known as stiff systems
We present a new class of collocation methods for numerical integration of Ordinary differential equations (ODEs) by basing the derivation on their integral form
The results show that changing the quadrature rule for the left-hand side of differential Equation (4) had a minuscule effect on the results
Summary
A special type of initial value problems, treated in this work, involves highly oscillatory systems and problems known as stiff systems. Ordinary differential equations (ODEs) describing these systems are ill-conditioned in a computational sense and challenging to solve [1]. For such problems, implicit Runge–Kutta (RK) methods are a convenient and frequent choice. A subset of implicit RK methods involves collocation methods. They yield a continuous approximation which makes them suitable for problems where globally continuously differentiable functions are required [5]. Despite being known for decades, the methods are still under continuous development. Publications examine the application of the methods to new problems, the derivation of new strategies, or the development of new procedures for solving nonlinear equations [3]. Three basic types of implicit RK methods, each based on different Gauss–Legendre-type quadrature formulae, are Gauss–Legendre quadrature formulae, Gauss–Radau quadrature formulae, and Gauss–Lobatto quadrature formulae [6]
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